Recovery and Rigidity in a Regular Stochastic Block Model

Abstract

The stochastic block model is a natural model for studying community detection in random networks. Its clustering properties have been extensively studied in the statistics, physics and computer science literature. Recently this area has experienced major mathematical breakthroughs, particularly for the binary (two-community) version, see [24, 25, 20]. In this paper, we introduce a variant of the binary model which we call the regular stochastic block model (RSBM). We prove rigidity of this model by showing that with high probability an exact recovery of the community structure is possible. Spectral methods exhibit a regime where this can be done efficiently. Moreover we also prove that, in this setting, any suitably good partial recovery can be bootstrapped to obtain a full recovery of the communities. Definition of the model and main resultsThe stochastic block model (SBM) is a classical clusterexhibiting random graph model that has been extensively studied, both empirically and rigorously, across numerous fields. In its simplest form, the SBM is a model of random graphs on 2n nodes with two equalsized clusters A and B such that |A| = |B| = n and A ∩ B = ∅. Edges between various pairs of vertices appear independently with probability p = p n if the two vertices belong to the same cluster and with probability q = q n otherwise. Thus, for any vertex, the expected number of same-class neighbors is a := a n := p(n − 1) ∼ pn, and the expected number of across-class neighbors is b := b n := qn.Given a realization of the graph, the broad goal is to determine whether it is possible (with high probability) to find the partition A, B; and if the answer is * Department of Mathematics, University of Washington, gerandy@math.washington.edu † Department of Mathematics, University of Washington, dumitriu@math.washington.edu ‡ Department of Mathematics, University of Washington, sganguly@math.washington.edu § Department of Mathematics, University of Washington, hoffman@math.washington.edu ¶ International University, National University Hochiminh City, tvlinh@hcmiu.edu.vn yes, whether it is possible to do so using an efficient algorithm. Otherwise, the best one can hope for is the existence of an algorithm that will output a partition which is highly (or at least positively) correlated with the underlying cluster. To this end, consider the space M of all algorithms which take as input a finite graph on 2n vertices and output a partition of the vertex set into two sets. Informally, we say that an algorithm in M allows for weak recovery if, with probability going to 1 as n goes to infinity, it outputs a partition (A ′ , B ′ ) such that |A∆A ′ | + |B∆B ′ | = o(n) (here ∆ denotes the symmetric difference). We say that an algorithm allows for strong recovery if, with probability going to 1 as n goes to infinity, it outputs the partition (A, B). Finally, an algorithm in M will be called efficient if its run time is polynomial in n.The problem of community detection described above is closely related to the min-bisection prob...

The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime

“…e distribution of ϑ(G) for the Erdős-Rényi graph G = G(n, p) and the random dregular graph G = G n,d were studied in [20]. In particular, that work showed that when d is sufficiently 1 ese models are not to be confused with a stricter model, where for some constants qrs each vertex in group r has exactly qrs neighbors in group s [18,23,53,15]. Our model only constrains the total number of edges within or between groups.…”

Spectral partitioning in equitable graphs

“…In SBM there exists a sharp transition in the assortativity parameter, first conjectured in [31] and later proved rigorously in a series of works that demonstrate both that asymptotically (i) below such threshold, recovery is information theoretically impossible [19] while (ii) above, an efficient algorithm finds a partition with positive overlap with the original one [20,21]. In this work, it is shown that the regularity condition, as also found in [13], substantially change the detectability properties of the ensemble.…”

“…This class of random graph models has been first analyzed in [14] in a dense approximation, in [15] for the first time under the name of equitable graphs and under the name microcanonical stochastic block model in [17]. In [13] it has been shown that equitable graphs with two-equallysized communities have a unique partition almost surely in the large size limit, that an efficient algorithm can be derived in a large region of the ensemble's parameters, and that full recovery of communities can be obtained starting from a group assignment with an extensive overlap with the original partition. In this paper, spectral theory of random graphs is used to disentangle noise and signal in equitable graphs, and an efficient algorithm for full recovery of communities is proposed.…”

The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences

“…Arguably the most popular model for community detection is the stochastic block model (SBM), where any two vertices within the same cluster or across different clusters are connected by an edge with probability p or q, respectively. The asymptotic limits for both exact and partial recovery have been extensively studied [1,2,10,26,38,39,57,60]. We note, however, that the primary focus of community detection lies in the sparse regime p; q 1=n or logarithmic sparse regime (i.e., p; q log n=n), which is in contrast to the joint alignment problem in which the measurements are often considerably denser.…”